Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted $Sp(2 n, F)$ and $Sp(n)$ for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by $\mathrm {USp} (n)$ . Many authors prefer slightly different notations, usually differing by factors of $2$ . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group $Sp(2 n, C)$ is denoted $C n$ , and $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension $n$ .

The name "symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".

The metaplectic group is a double cover of the symplectic group over R; it has analogues over other local fields, finite fields, and adele rings.

$Sp(2 n, F)$

The symplectic group is a classical group defined as the set of linear transformations of a $2 n$ -dimensional vector space over the field $F$ which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space $V$ is denoted $Sp(V)$ . Upon fixing a basis for $V$ , the symplectic group becomes the group of $2 n \times 2 n$ symplectic matrices, with entries in $F$ , under the operation of matrix multiplication. This group is denoted either $Sp(2 n, F)$ or $Sp(n, F)$ . If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then

\operatorname {Sp} (2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \},

where M^T is the transpose of M. Often Ω is defined to be

\Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},

where I_n is the identity matrix. In this case, $Sp(2 n, F)$ can be expressed as those block matrices $({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})$ , where $A,B,C,D\in M_{n\times n}(F)$ , satisfying the three equations:

{\begin{aligned}-C^{\mathrm {T} }A+A^{\mathrm {T} }C&=0,\\-C^{\mathrm {T} }B+A^{\mathrm {T} }D&=I_{n},\\-D^{\mathrm {T} }B+B^{\mathrm {T} }D&=0.\end{aligned}}

Since all symplectic matrices have determinant $1$ , the symplectic group is a subgroup of the special linear group $SL(2 n, F)$ . When $n = 1$ , the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that $Sp(2, F) = SL(2, F)$ . For $n > 1$ , there are additional conditions, i.e. $Sp(2 n, F)$ is then a proper subgroup of $SL(2 n, F)$ .

Typically, the field $F$ is the field of real numbers $R$ or complex numbers $C$ . In these cases $Sp(2 n, F)$ is a real or complex Lie group of real or complex dimension $n (2 n + 1)$ , respectively. These groups are connected but non-compact.

The center of $Sp(2 n, F)$ consists of the matrices $I 2 n$ and $- I 2 n$ as long as the characteristic of the field is not $2$ .^[1] Since the center of $Sp(2 n, F)$ is discrete and its quotient modulo the center is a simple group, $Sp(2 n, F)$ is considered a simple Lie group.

The real rank of the corresponding Lie algebra, and hence of the Lie group $Sp(2 n, F)$ , is $n$ .

The Lie algebra of $Sp(2 n, F)$ is the set

{\mathfrak {sp}}(2n,F)=\{X\in M_{2n\times 2n}(F):\Omega X+X^{\mathrm {T} }\Omega =0\},

equipped with the commutator as its Lie bracket.^[2] For the standard skew-symmetric bilinear form $\Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})$ , this Lie algebra is the set of all block matrices $({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})$ subject to the conditions

{\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}

$Sp(2 n, C)$

The symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group.

$Sp(2 n, R)$

$Sp(n, C)$ is the complexification of the real group $Sp(2 n, R)$ . $Sp(2 n, R)$ is a real, non-compact, connected, simple Lie group.^[3] It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra.

Some further properties of $Sp(2 n, R)$ :

The exponential map from the Lie algebra $sp (2 n, R)$ to the group $Sp(2 n, R)$ is not surjective. However, any element of the group can be represented as the product of two exponentials.^[4] In other words,

\forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.

For all $S$ in $Sp(2 n, R)$ :

S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.

The matrix

D

is positive-definite and diagonal. The set of such

Z

s forms a non-compact subgroup of

Sp(2 n, R)

whereas

U(n)

forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.^[5] Further symplectic matrix properties can be found on that Wikipedia page.

As a Lie group, $Sp(2 n, R)$ has a manifold structure. The manifold for $Sp(2 n, R)$ is diffeomorphic to the Cartesian product of the unitary group $U(n)$ with a vector space of dimension $n (n +1)$ .^[6]

Infinitesimal generators

The members of the symplectic Lie algebra $sp (2 n, F)$ are the Hamiltonian matrices.

These are matrices, $Q$ such that

$Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}$

where $B$ and $C$ are symmetric matrices. See classical group for a derivation.

Example of symplectic matrices

For $Sp(2, R)$ , the group of $2 \times 2$ matrices with determinant $1$ , the three symplectic $(0, 1)$ -matrices are:^[7]

${\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.$

Sp(2n, R)

It turns out that $\operatorname {Sp} (2n,\mathbf {R} )$ can have a fairly explicit description using generators. If we let $\operatorname {Sym} (n)$ denote the symmetric $n\times n$ matrices, then $\operatorname {Sp} (2n,\mathbf {R} )$ is generated by $D(n)\cup N(n)\cup \{\Omega \},$ where

${\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}$

are subgroups of $\operatorname {Sp} (2n,\mathbf {R} )$ ^[8]^{pg 173}^[9]^{pg 2}.

Relationship with symplectic geometry

Symplectic geometry is the study of symplectic manifolds. The tangent space at any point on a symplectic manifold is a symplectic vector space.^[10] As noted earlier, structure preserving transformations of a symplectic vector space form a group and this group is $Sp(2 n, F)$ , depending on the dimension of the space and the field over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under an action of the symplectic group is thus, in a sense, a linearised version of a symplectomorphism which is a more general structure preserving transformation on a symplectic manifold.

$Sp(n)$

The compact symplectic group^[11] $Sp(n)$ is the intersection of $Sp(2 n, C)$ with the $2n\times 2n$ unitary group:

\operatorname {Sp} (n):=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {U} (2n)=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {SU} (2n).

It is sometimes written as $USp(2 n)$ . Alternatively, $Sp(n)$ can be described as the subgroup of $GL(n, H)$ (invertible quaternionic matrices) that preserves the standard hermitian form on $H n$ :

\langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.

That is, $Sp(n)$ is just the quaternionic unitary group, $U(n, H)$ .^[12] Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of norm $1$ , equivalent to $SU(2)$ and topologically a $3$ -sphere $S 3$ .

Note that $Sp(n)$ is not a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric $H$ -bilinear form on $H n$ : there is no such form except the zero form. Rather, it is isomorphic to a subgroup of $Sp(2 n, C)$ , and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of $Sp(n)$ is the compact real form of the complex symplectic Lie algebra $sp (2 n, C)$ .

$Sp(n)$ is a real Lie group with (real) dimension $n (2 n + 1)$ . It is compact and simply connected.^[13]

The Lie algebra of $Sp(n)$ is given by the quaternionic skew-Hermitian matrices, the set of $n -by- n$ quaternionic matrices that satisfy

A+A^{\dagger }=0

where $A †$ is the conjugate transpose of $A$ (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

Important subgroups

Some main subgroups are:

\operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)

\operatorname {Sp} (n)\supset \operatorname {U} (n)

\operatorname {Sp} (2)\supset \operatorname {O} (4)

Conversely it is itself a subgroup of some other groups:

\operatorname {SU} (2n)\supset \operatorname {Sp} (n)

\operatorname {F} _{4}\supset \operatorname {Sp} (4)

\operatorname {G} _{2}\supset \operatorname {Sp} (1)

There are also the isomorphisms of the Lie algebras $sp (2) = so (5)$ and $sp (1) = so (3) = su (2)$ .

Relationship between the symplectic groups

Every complex, semisimple Lie algebra has a split real form and a compact real form; the former is called a complexification of the latter two.

The Lie algebra of $Sp(2 n, C)$ is semisimple and is denoted $sp (2 n, C)$ . Its split real form is $sp (2 n, R)$ and its compact real form is $sp (n)$ . These correspond to the Lie groups $Sp(2 n, R)$ and $Sp(n)$ respectively.

The algebras, $sp (p, n - p)$ , which are the Lie algebras of $Sp(p, n - p)$ , are the indefinite signature equivalent to the compact form.

Physical significance

Classical mechanics

The non-compact symplectic group $Sp(2 n, R)$ comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of $n$ particles, evolving under Hamilton's equations whose position in phase space at a given time is denoted by the vector of canonical coordinates,

\mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.

The elements of the group $Sp(2 n, R)$ are, in a certain sense, canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations.^[14]^[15] If

\mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }

are new canonical coordinates, then, with a dot denoting time derivative,

{\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},

where

M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )

for all $t$ and all $z$ in phase space.^[16]

For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics on that manifold. The coordinates $q^{i}$ live on the underlying manifold, and the momenta $p_{i}$ live in the cotangent bundle. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is $H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}$ where $g^{ij}$ is the inverse of the metric tensor $g_{ij}$ on the Riemannian manifold.^[17]^[15] In fact, the cotangent bundle of any smooth manifold can be a given a symplectic structure in a canonical way, with the symplectic form defined as the exterior derivative of the tautological one-form.^[18]

Quantum mechanics

Consider a system of $n$ particles whose quantum state encodes its position and momentum. These coordinates are continuous variables and hence the Hilbert space, in which the state lives, is infinite-dimensional. This often makes the analysis of this situation tricky. An alternative approach is to consider the evolution of the position and momentum operators under the Heisenberg equation in phase space.

Construct a vector of canonical coordinates,

\mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.

The canonical commutation relation can be expressed simply as

[\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega

where

\Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}

and $I n$ is the $n \times n$ identity matrix.

Many physical situations only require quadratic Hamiltonians, i.e. Hamiltonians of the form

{\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}}

where $K$ is a $2 n \times 2 n$ real, symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the Heisenberg equation as

{\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}}

The solution to this equation must preserve the canonical commutation relation. It can be shown that the time evolution of this system is equivalent to an action of the real symplectic group, $Sp(2 n, R)$ , on the phase space.

Notes

^ "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.
^ Hall 2015 Prop. 3.25
^ "Is the symplectic group Sp(2n, R) simple?", Stack Exchange Retrieved on 14 December 2014.
^ "Is the exponential map for Sp(2n, R) surjective?", Stack Exchange Retrieved on 5 December 2014.
^ "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso", Retrieved on 30 January 2015.
^ "Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.
^ Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012
^ Gerald B. Folland. (2016). Harmonic analysis in phase space. Princeton: Princeton Univ Press. p. 173. ISBN 978-1-4008-8242-7. OCLC 945482850.
^ Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN 978-3-540-33421-7. OCLC 262692314.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^ "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.
^ Hall 2015 Section 1.2.8
^ Hall 2015 p. 14
^ Hall 2015 Prop. 13.12
^ Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.
^ ^a ^b Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X
^ Goldstein 1980, Section 9.3
^ Jurgen Jost, (1992) Riemannian Geometry and Geometric Analysis, Springer.
^ da Silva, Ana Cannas (2008). Lectures on Symplectic Geometry. Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.

References

Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60 (second ed.), Springer-Verlag, ISBN 0-387-96890-3
Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
Fulton, W.; Harris, J. (1991), Representation Theory, A first Course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, ISBN 978-0-387-97495-8.
Goldstein, H. (1980) [1950]. "Chapter 7". Classical Mechanics (2nd ed.). Reading MA: Addison-Wesley. ISBN 0-201-02918-9.
Lee, J. M. (2003), Introduction to Smooth manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, ISBN 0-387-95448-1
Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information", arXiv:quant-ph/0503237.

[1] "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.

[2] Hall 2015 Prop. 3.25

[3] "Is the symplectic group Sp(2n, R) simple?", Stack Exchange Retrieved on 14 December 2014.

[4] "Is the exponential map for Sp(2n, R) surjective?", Stack Exchange Retrieved on 5 December 2014.

[5] "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso", Retrieved on 30 January 2015.

[6] "Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.

[7] Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012

[8] Gerald B. Folland. (2016). Harmonic analysis in phase space. Princeton: Princeton Univ Press. p. 173. ISBN 978-1-4008-8242-7. OCLC 945482850.

[9] Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN 978-3-540-33421-7. OCLC 262692314.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

[10] "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.

[11] Hall 2015 Section 1.2.8

[12] Hall 2015 p. 14

[13] Hall 2015 Prop. 13.12

[14] Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.

[A&M-15] Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X

[16] Goldstein 1980, Section 9.3

[17] Jurgen Jost, (1992) Riemannian Geometry and Geometric Analysis, Springer.

[18] Silva, Ana Cannas (2008). Lectures on Symplectic Geometry. Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.

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